Terence Tao

• 1. Introduction

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  Terence Tao scored 760 on the math section of the SAT when he was only eight years old. Only two other children in the Johns Hopkins Study of Exceptional Talent program have ever reached 700 or more at that age. That boy grew up to be an Australian and American mathematician, and he now teaches at the University of California, Los Angeles. His career has crossed so many branches of mathematics that experts struggle to place it in any single one. This documentary follows the path from a gifted child to a highly cited, widely collaborative and heavily decorated mathematician. Along the way it asks what he actually proved and how he works with so many partners. It also asks why a pure mathematician has written about research budgets.

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• 2. Adelaide to Princeton

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  Billy Tao and Grace Leong met as students at the University of Hong Kong. Billy, a paediatrician born in Shanghai, earned his medical degree there in 1969. Grace, born in Hong Kong, graduated with first-class honours in mathematics and physics. She went on to teach both subjects at a secondary school. The couple emigrated to Australia in 1972, and their son Terence was born on the 17th of July 1975. He grew up in Adelaide, South Australia, alongside two brothers named Trevor and Nigel. Both brothers later represented Australia at the International Mathematical Olympiad.

  Terence skipped five grades at school. By the age of nine he was attending university-level mathematics courses. Julian Stanley, who directed the Study of Mathematically Precocious Youth, said Tao had the strongest mathematical reasoning ability he had found in years of intensive searching.

  At the age of ten, Tao became the youngest person ever to compete in the International Mathematical Olympiad. He won a bronze medal in 1986, a silver in 1987 and a gold in 1988. No one younger has ever won any of those three medals. At fourteen he attended the Research Science Institute, a summer program for secondary students.

  In 1991, at sixteen, Tao received both his bachelor's and master's degrees from Flinders University, working under Garth Gaudry. In 1992 he won a postgraduate Fulbright Scholarship to study mathematics at Princeton University. He remained a graduate student there until 1996 under Elias Stein, and he received his PhD at twenty-one. In 1996 he joined the faculty of the University of California, Los Angeles. In 1999, at twenty-four, he became a full professor, the youngest person that university has ever appointed to the rank. Today he holds the James and Carol Collins Chair in the College of Letters and Sciences.

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• 3. Primes in Progression

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  Three, seven and eleven form a progression of prime numbers with a spacing of four. The next number in that sequence, fifteen, is not prime, so the pattern stops. In 2004 Tao and the British mathematician Ben Green proved that such progressions of primes exist at any length you choose. Their result is called the Green-Tao theorem, and amateur and professional mathematicians alike know it well. The New York Times described it as the solution to a problem related to the Twin Prime Conjecture.

  Endre Szemerédi proved a theorem in 1975 about arithmetic progressions inside certain sets of integers. Green and Tao showed that a "transference principle" could stretch that theorem to further sets of integers. The primes then become a special case, though proving that they meet the required conditions is not trivial.

  In 2010 Green and Tao gave a multilinear extension of Dirichlet's celebrated theorem on arithmetic progressions. Their first proof was incomplete, because it depended on conjectures that nobody had yet proved. Those conjectures were later proved in work by Green, Tao and Tamar Ziegler.

  Also in 2004, Tao joined Jean Bourgain and Nets Katz to study sums and products inside subsets of finite fields of prime order. Such a field has no nontrivial subrings, and the trio turned that fact into a quantitative statement. Any subset must generate a large number of sums and products, measured against the size of the field and the subset. Bourgain, Alexey Glibichuk and Sergei Konyagin later improved the result.

  In 2012 Green and Tao announced proofs of the "orchard-planting problem." It asks for the maximum number of lines through exactly three points in a set of n points in the plane, when the points do not all lie on one line.

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• 4. Waves and Harmonics

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  Between 2001 and 2010, Tao collaborated with James Colliander, Markus Keel, Gigliola Staffilani and Hideo Takaoka on dispersive partial differential equations. The group found many new results about Schrödinger and KdV equations, often concerning well-posedness of weak solutions. Well-posedness asks, loosely, whether solutions to an equation exist and behave predictably. Michael Christ, Colliander and Tao built on methods of Carlos Kenig, Gustavo Ponce and Luis Vega to show ill-posedness for certain equations at low Sobolev exponents. In many cases those results fit neatly against well-posedness results at larger exponents.

  One notable result from the five-person group established long-time existence and scattering theory for a power-law Schrödinger equation in three dimensions. Their method relied on the scale-invariance of the simple power law. Tao later extended it with Monica Vișan and Xiaoyi Zhang to cases where scale-invariance breaks. Rowan Killip, Tao and Vișan then made progress on the two-dimensional problem under radial symmetry.

  A 2001 article by Tao tackled the wave maps equation with a two-dimensional domain and a spherical range. He built on earlier innovations by Daniel Tataru. Tao proved global well-posedness for solutions with sufficiently small initial data, where smallness is measured against the critical Sobolev norm. That setting usually demands sophisticated techniques. He later adapted part of this work to the Benjamin-Ono equation, and Alexandru Ionescu and Kenig used his methods to obtain improved results.

  Bent Fuglede proposed his conjecture in the 1970s, linking tiling to Fourier bases for Euclidean domains. Tao resolved it in the negative for dimensions greater than five. He did so by building a simple counterexample in the setting of finite groups.

  With Ana Vargas and Luis Vega, Tao made foundational contributions to the bilinear restriction problem, establishing new exponents and linking it to the linear restriction problem. They found parallel results for the bilinear Kakeya problem. Tao, Jonathan Bennett and Anthony Carbery then developed the multilinear setting. Bourgain and Larry Guth made extensive use of that work in estimates for oscillatory integral operators.

  In 2016 Tao constructed a variant of the Navier-Stokes equations whose solutions behave irregularly in finite time. Because his system resembles the real equations, any positive answer to the Navier-Stokes existence and smoothness problem must respect their specific nonlinear structure. Certain earlier proposed answers therefore could not be legitimate. Tao speculated that the real equations might simulate a Turing complete system, which could allow a negative resolution. As of 2025 that idea remains conjectural.

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• 5. Signals and Randomness

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  Emmanuel Candès, Justin Romberg and Tao studied a puzzle in a field called compressed sensing. The puzzle involves an underdetermined linear system, where the goal is a solution with as few nonzero entries as possible. That property is called sparsity. Their results identify settings where an easier convex optimisation problem correctly solves a problem that looks computationally hopeless. David Donoho studied similar questions around the same time from the viewpoint of high-dimensional geometry.

  Striking numerical experiments motivated the trio to start with the case where the matrix comes from the discrete Fourier transform. Candès and Tao then abstracted the problem and introduced the notion of a "restricted linear isometry." Such a matrix stays close to an isometry when restricted to certain subspaces. They showed this property suffices for exact or optimally approximate recovery of sufficiently sparse solutions. Their first proofs used convex duality, and working with Romberg they simplified them to linear algebra and elementary harmonic analysis. Drawing on random matrix theory, they also showed that a large number of matrices from the Gaussian ensemble have the property.

  In 2007 Candès and Tao introduced a statistical estimator for linear regression called the "Dantzig selector." Trevor Hastie, Robert Tibshirani and Jerome H. Friedman concluded that it is "somewhat unsatisfactory" in a number of cases. It nonetheless remains of significant interest in the statistical literature. In 2009 Candès and Benjamin Recht tackled recovering a low-rank matrix from only a few of its entries, and Candès and Tao developed further results in 2010.

  Eugene Wigner began the study of random matrices in the 1950s, proving a "semicircle law" for the eigenvalues of hermitian and symmetric matrices. In 2010 Tao and Van Vu made a major contribution for non-symmetric matrices. They showed that the eigenvalues of a large random matrix scatter uniformly across a disk, for any fixed distribution with expectation 0 and standard deviation 1. This proved the long-conjectured circular law, which others had established only in weaker forms. Their key tool was a "universality principle," which says the eigenvalue distribution depends only on the average and standard deviation.

  In 2011 Tao and Van Vu established a "four moment theorem" for random hermitian matrices. Similar results appeared around the same time from László Erdős, Horng-Tzer Yau and Jun Yin.

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• 6. Collaboration and Conjectures

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  By 2006 Tao had co-authored publications with more than 30 other researchers. By October 2015 the count had reached 68 co-authors. The New Scientist reported that mathematicians now compete to interest him in their problems, and that he is becoming a kind of Mr. Fix-it for frustrated researchers. Charles Fefferman, a professor of mathematics at Princeton University, put it this way: "If you're stuck on a problem, then one way out is to interest Terence Tao."

  The British mathematician and Fields medalist Timothy Gowers wrote about the breadth of Tao's knowledge. He listed topics from partial differential equations and analytic number theory to quantum mechanics and image processing. Some are fields where Tao has made fundamental contributions, and others are fields he does not officially work in. Gowers added: "How he does all this, as well as writing papers and books at a prodigious rate, is a complete mystery." He noted that David Hilbert is said to have been the last person to know all of mathematics. In Tao's case, gaps in knowledge tend to be filled within a year.

  In 2015 Tao resolved the Erdős discrepancy problem, using entropy estimates from analytic number theory. In 2018 he worked with Brad Rodgers to show that the de Bruijn-Newman constant is nonnegative. Its nonpositivity is equivalent to the Riemann hypothesis. In 2019 he proved that almost all Collatz orbits attain almost bounded values, a probabilistic step toward the Collatz conjecture. In 2020 he proved Sendov's conjecture, about the locations of roots and critical points of a complex polynomial, for polynomials of sufficiently high degree. In 2024 and 2025 he solved Erdős problems 121, 442, 135, 685, 69 and 1102.

  Tao has an Erdős number of 2. As of 2026 he had published about four hundred thirty articles and nineteen books.

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• 7. Honours and Reluctance

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  In 2006 Tao received the Fields Medal for contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory. He was the first Australian and the first UCLA faculty member to win it, and one of the youngest recipients. The same year he became a MacArthur Fellow, and in 2007 he was elected a Fellow of the Royal Society. He is also a member of the American Academy of Arts and Sciences and the American Philosophical Society, among other academies.

  In 2014 Tao was named one of the inaugural Breakthrough Prize laureates in mathematics. He felt unqualified for it and argued, without success, that the prize money should be spread among more researchers. After receiving the money, he used part of it to establish fellowships for students. That year he also received the Royal Medal. He returned to the program that had nurtured him when the Johns Hopkins Center for Gifted and Talented Youth gave him a Distinguished Alumni Honor. The audience of 979 attendees in the 8th and 9th grades were in the same program he had graduated from.

  In 2019 the Riemann International School of Mathematics at the University of Insubria chose Tao as the first winner of the Riemann Prize. In 2020 he shared the Princess of Asturias Award for Technical and Scientific Research with Emmanuel Candès, for their work on compressed sensing. Advance named him Global Australian of the Year in 2022.

  In 2021 President Joe Biden announced Tao as one of 30 members of the President's Council of Advisors on Science and Technology. The council brings together some of America's most distinguished leaders in science and technology.

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• 8. Funding and Home

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  On the 18th of August 2025, Tao wrote an article objecting to President Donald Trump's policies cutting research funding. The cuts directly touched his own work. The National Science Foundation suspended two of his research grants as part of a broader federal action affecting UCLA. One grant supported his research at UCLA. The other supported the university's Institute for Pure and Applied Mathematics, where Tao oversees research projects.

  Tao argued that the cuts harm the recruitment of academic talent and slow the advance of research. His own field is pure mathematics, yet it could lay foundations for applied work in areas such as cryptography and cybersecurity. He pointed to his earlier collaborative research in signal processing, which greatly accelerated MRI scan speed.

  Tao is married to Laura Tao, an electrical engineer at NASA's Jet Propulsion Laboratory. They live in Los Angeles, California, and have two children. Tao speaks Cantonese but cannot write Chinese.

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